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Nauenberg 226 7.11.2004 11:53 Uhr Seite 1 Nauenberg 226 7.11.2004 11:53 Uhr Seite 1 Phys. perspect. 7 (2005) xxx–yyy 1422-6944/05/010xxx–31 DOI 10.1007/s00016-004-0226-? Robert Hooke’s Seminal Contribution to Orbital Dynamics Michael Nauenberg* During the second half of the seventeenth century, the outstanding problem in astronomy was to understand the physical basis for Kepler’s laws describing the observed orbital motion of a planet around the Sun. Robert Hooke (1635–1703) proposed in the middle 1660s that a planet’s motion is determined by compounding its tangential velocity with its radial velocity as impressed by the gravitational attraction of the Sun, and he described his physical concept to Isaac Newton (1642–1726) in correspondence in 1679. Newton denied having heard of Hooke’s novel concept of orbital motion,but shortly after their correspondence he implemented it by a geometric construction from which he deduced the physical origin of Kepler’s area law, which later became Proposition I, Book I, of his Principia in 1687. Three years earlier, Newton had deposited a preliminary draft of it, his De Motu Corporum in Gyrum (On the Motion of Bodies), at the Royal Society of Lon- don, which Hooke apparently was able to examine a few months later, since shortly thereafter he applied Newton’s construction in a novel way to obtain the path of a body under the action of an attractive central force that varies linearly with the distance from its center of motion (Hooke’s law). I show that Hooke’s construction corresponds to Newton’s for his proof of Kepler’s area law in his De Motu. Hooke’s understanding of planetary motion was based on his observations with mechanical analogs.I repeated two of his experiments and demonstrated the accuracy of his obser- vations. My results thus cast new light on the significance of Hooke’s contributions to the devel- opment of orbital dynamics, which in the past have either been neglected or misunderstood. Key words: Robert Hooke; Isaac Newton; astronomy; orbital motion. Introduction One of the most fascinating questions in the history of science concerns the role that Robert Hooke (1635–1703) played in the development of dynamics and the theory of gravitation during the seventeenth century, which culminated in Isaac Newton’s mas- terpiece, the Principia in 1687.1 Hooke was one of the most prolific and inventive sci- entists of all times, and he made fundamental contributions to virtual all branches of science;2 one third of the fifteen volumes of Robert T. Gunther’s Early Science in Oxford are dedicated to his work.3 Despite Hooke’s profound influence, however, particularly on Newton’s develop- ment of orbital dynamics, he was nearly completely forgotten after his death in 1703, * Michael Nauenberg is Professor Emeritus of Physics at the University of California,Santa Cruz. His primary research has been in theoretical physics, but he also has written several articles and co-edited a book on the historical development of dynamics by Huygens, Newton, and Hooke. 1 Nauenberg 226 7.11.2004 11:53 Uhr Seite 2 2 Michael Nauenberg Phys. perspect. and remained unknown until around the turn of the 20th century.* Ernst Mach, in his influential book of 1883, The Science of Mechanics, devoted only a few lines to Hooke, although he perceptively stated that “Hooke really approached nearest to Newton’s conception, though he never completely reached the latter’s altitude of view.”4 Ten years later, W.W. Rouse Ball published some of the correspondence between Hooke and Newton (1642–1726),5 and the subsequent discovery of two additional letters of Hooke, which were published by Jean Pelseneer and Alexandre Koyré,6 initiated a reappraisal of Hooke’s contributions to mechanics that continues to the present time.7 René Dugas, in contrast to Mach, recognized Hooke’s crucial role in his book of 1958, Mechanics in the Seventeenth Century,8 but some recent accounts of the development of mechanics still ignore Hooke’s significant contributions. Most physicists and mathe- maticians still remain unaware of them, as can be seen by reading modern textbooks or journals that cover classical mechanics, where Hooke is mentioned only in connec- tion with his eponymous law of elasticity.9 During the past few years, however, a number of books and articles have appeared that describe Hooke’s many contributions to science, and also his major role in the reconstruction of London after the Great Fire of 1666.10 There now appears to be con- sensus among historians of science that Hooke’s physical explanation for the orbital motion of planets – “compounding the celestiall motions of the planetts of a direct motion by the tangent & an attractive motion towards the centrall body” 11 – had an influence on Newton’s work that culminated in the publication of his Principia. Hooke communicated his ideas on orbital motion to Newton in a letter in 1679 and, according to one of Newton’s most outstanding biographers, Richard S. Westfall: Newton’s papers reveal no similar understanding of circular motion before this let- ter. Every time he had considered it, he had spoken of a tendency to recede from the center, what [Christiaan] Huygens [1629–1695] called centrifugal force; and like oth- ers who spoke in such terms, he had looked upon circular motion as a state of equi- librium between two equal and opposing forces, one away from the center and one toward it. Hooke’s statement treated circular motion as a disequilibrium in which an unbalanced force deflects a body that would otherwise continue in a straight line. It was not an inconsiderable lesson for Newton to learn.12 But there is considerable confusion in the literature as to exactly what the “lesson” was that Newton learned from Hooke. In a letter to Edmond Halley (1656–1742) on June 20, 1679,13 Newton vehemently denied that he had learned anything form Hooke, although he admitted, in one of his unpublished autobiographical manuscripts, that: In the end of the year 1679 in answer to a Letter from Dr Hook … I found now that whatsoever was the law of the forces wch kept the Planets in their Orbs, the areas described by a Radius drawn from them to the Sun would be proportional to the times in wch they were described. And by the help of these two Propositions I found that their Orbs would be such Ellipses as Kepler had described.14 * Symtomatic of the neglect of Hooke and his legacy is that he is virtually alone among the great scientists of the past for whom no extant portrait exists. Nauenberg 226 7.11.2004 11:53 Uhr Seite 3 Vol. 7 (2005) Hooke’s Contribution to Orbital Dynamics 3 To understand the influence of Hooke’s concept of orbital motion on Newton, we must know how Newton viewed orbital dynamics prior to Hooke’s letter to him in 1679. Newtonian scholars commonly conclude that Newton’s crucial step thereafter was to switch from a traditional view of circular motion as giving rise to a centrifugal force or tendency to recede from the center, to the concept of a centripetal force directed toward the center.* As I have shown, however, this is misleading in view of Newton’s already profound albeit incomplete understanding of orbital motion at that time.15 In the following sections, I first describe Newton’s early development of orbital dynamics as based on his “Waste Book” of 1664 and on his letter of December 13, 1679, to Hooke. I then present Hooke’s formulation of the physical principles of orbital motion for central forces and analyze his recently discovered diagram of 1685 for the motion of a body under the action of a central force that varies linearly with the dis- tance.16 I show graphically that it implements his dynamical principles in a way very similar to Newton’s description of his proof of Kepler’s area law in his De Motu Cor- porum in Gyrum (On the Motion of Bodies) of 1684,17 the preliminary draft of his Prin- cipia. I next present further background on the development of Hooke’s physical ideas, which were based on mechanical analogs. I repeated Hooke’s experiments on a coni- cal pendulum and on a ball rolling inside an inverted cone, finding both to agree with Hooke’s observations. I close with a summary and conclusions. Newton’s Theory of Orbital Motion Prior to Hooke’s Letter of 1679 René Descartes (1596–1650) illustrated the traditional view of circular motion in 1644 by considering a stone rotating in a sling (figure 1).18 If the stone is freed at point A, then it would move along the tangent to the circle from point A to point C, but instead the sling constrains it to move along a circular path to point B. The “tendency” of the stone to recede radially is felt by the person’s hand at the center, since the stone exerts a force that depends on its weight and velocity and on the length of the sling. The mag- nitude of this force was not known quantitatively until Huygens and somewhat later Newton showed that it is proportional to the radial acceleration,19 which is equal to the square of the velocity of the stone divided by the length of the sling. Prior to 1679, Newton based his description of the orbital motion of a body under the action of a central force on a generalization of the properties of circular motion,20 compounding its orbital velocity along the tangent with a change of velocity perpen- dicular to this direction, whereas Hooke considered its total change in velocity direct- ed toward the center of force.
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